Name: Date: Lesson 5 Section 5.1: Linear Functions vs. Exponential Functions 1. Complete the table below. Function Linear or Exponential? Linear: Increasing or Decreasing? Exponential: Growth or Decay? Linear: find the slope Exponential: find the base Identify the Vertical Intercept as an Ordered Pair.! = 2! + 4 Linear Increasing Slope = 2 (0,4)!(!) = 3 2!!(!) = 1.5! 2!(!) = 100 1.2!!(!) = 1.8! + 32!(!) = 1000 0.75! Page 189
2. For the following three linear functions, identify the vertical intercept, calculate the slope and then write the equation for the function in f(x) = mx + b form. a) x f(x) -1 3 0 2 1 1 b) f(x) = {(-2,2), (0,3), (2,4)} c) f(x) 3. For the following three exponential functions, identify the initial value (a), calculate the base (b), and then write the equation for the function in f(x) = ab x form. a) x f(x) 0 4 1 8 2 16 b) f(x) = {(0, 2), (1, 4.2), (2, 8.82)} c) 4-4 4 f(x) Page 190
4. Determine if each data set is linear or exponential, and write the formula for each. Show complete work. a) x 2 1 0 1 2 3 4 f(x).04.2 1 5 25 125 625 b) x 2 1 0 1 2 3 4 f(x) 1.375.5.375 1.25 2.125 3 3.875 c) x 2 1 0 1 2 3 4 f(x) 3 5.5 8 10.5 13 15.5 18 d) x 2 1 0 1 2 3 4 f(x) 98.224 99.108 100 100.9 101.81 102.72 103.65 e) x 0 1 2 3 4 5 6 f(x) 2 4 8 16 32 64 128 Page 191
Section 5.2: Characteristics of Exponential Functions 5. Complete the table below for each exponential function. Initial Value (a) f(x) = 3.4(1.13) x g(x) = 42(0.62) x h(x) = 1000(1.03) x Base (b) Domain Range Horizontal Intercept Vertical Intercept Horizontal Asymptote Increasing or Decreasing Growth or Decay Page 192
Section 5.3: Solving Exponential Equations 6. Given!(!) = 50 1.25!, determine each of the following and show complete work. a) f(5) = b) f(50) = c) Find x when f(x) = 75 d) Find x when f(x) = 25 7. Given!! = 100 0.90!, determine each of the following and show complete work. a) f(3) = b) f(30)= c) Find x when f(x) = 25 d) Find x when f(x) = 50 8. Given!! = 25 3!, determine each of the following and show complete work. a) f(1) = b) f(3)= c) Find x when f(x) = 100 d) Find x when f(x) = 5000 Page 193
Section 5.4: Applications of Exponential Funcitons 9. The rabbit population in several counties is shown in the following table. Assume this growth is exponential. Let t = 0 represent the year 2006. Let a represent the initial population in 2006. Let b represent the ratio in population between the years 2006 and 2007. Rabbit Population Year Coconino Yavapai Harestew 2006 15000 8000 25000 2007 18000 12800 18750 2008 21600 20480 14063 2009 25920 32768 10547 a) Write the equation of the exponential mathematical model for each situation. Round any decimals to two places. Be sure your final result uses proper function notation. Use C(t) for Coconino, Y(t) for Yavapai and H(t) for Harestew. b) Use the models from part a) to forecast the rabbit population in 2012 for each county. Round to the nearest rabbit. Use proper function notation to represent each result. Page 194
c) Use the models from part a) to find the following. Show complete work. i. The Rabbit Population in Coconino County reaches 60,000. ii. The Rabbit Population in Yavapai Country reaches 340,000. iii. The Rabbit Population in Harestew falls below 5000. d) Which of the scenarios from part c) happened first? Explain your reasoning. Page 195
10. Assume you can invest $1000 at 5% Simple Interest or 4% Compound Interest (Annual). The equation for Simple Interest is modeled by: A = P + Prt. Compound Interest is modeled by A = P(1+r) t. The corresponding equations for these two types of interest are given below. S(t) = 1000 + 50t C(t) = 1000(1.04) t a) Complete the table for each function. t 1 5 10 15 20 S(t) C(t) b) What is the vertical intercept for each function and what does it represent in the context of this problem? c) Graph these two functions on the same graph. Plot and label their intersection. Use window Xmin=0, Xmax = 20, Ymin=1000, Ymax=2500. d) When would the two investments return the same amount? How much would they return? e) Which investment would you go with in the short term (less than 10 years)? Explain. Page 196
11. In 2010, the estimated population of Maricopa County was 3,817,117. By 2011, the population had grown to 3,880,244. a) Assuming that the growth is linear, construct a linear equation that expresses the population, P, of Maricopa County t years since 2010. b) Assuming that the growth is exponential, construct an exponential equation that expresses the population, P, of Maricopa County t years since 2010. c) Use the equation found in part a) to predict the population of Maricopa County in 2015. d) Use the equation found in part b) to predict the population of Maricopa County in 2015. Page 197
12. In each situation below, you will need to graph to find the solution to the equation using the INTERSECTION method described in this lesson. Fill in the missing information for each situation. Include a rough but accurate sketch of the graphs and intersection point. Mark and label the intersection. Round answers to two decimal places. a) Solve 25(1.25) x = 400 Solution: x = Xmin: Xmax: Ymin: Ymax: b) Solve 300(0.85) x = 80 Solution: x = Xmin: Xmax: Ymin: Ymax: c) Solve 300(0.85) x = 1700 Solution: x = Xmin: Xmax: Ymin: Ymax: Page 198
d) Solve 17.5(2.05) x = 1 Solution: x = Xmin: Xmax: Ymin: Ymax: e) Solve 2(1.01) x = 12 Solution: x = Xmin: Xmax: Ymin: Ymax: f) Solve 532(0.991) x = 100 Solution: x = Xmin: Xmax: Ymin: Ymax: Page 199
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